Proposition 4

If the integral of $\mathbf{f}$ over $I$ is absolutely convergent then it is convergent, and one has

Indeed, for every compact interval $J \subset I$ one has (II, p. 61, formula (16))

If the integral of the positive function $\| \mathbf{f}(x) \|$ is convergent, then for every $\varepsilon > 0$ there exists a compact interval $[\alpha,\ \beta]$ contained in $I$, such that, for every compact interval $[\lambda,\ \gamma]$ contained in $I$ and having no interior point in common with $[\alpha,\ \beta]$, one has $\int_{\lambda}^{\gamma} \| \mathbf{f}(t) \, dt \| \leq \varepsilon$ (II, p. 65, prop. 1); one deduces that $\left| \int_{\lambda}^{\gamma} \mathbf{f}(t)\,dt \right| \leq \varepsilon$, which shows convergence of the integral over $I$ (II, p. 16, prop. 1); on passing to the limit in (3) one deduces the inequality (2).